Non-Fermi liquid vs (topological) Mott insulator in electronic systems with quadratic band

touching in three dimensions

Igor Herbut (Simon Fraser University, Vancouver)

Lukas Janssen (SFU)Balazs Dora (Budapest) Roderich Moessner (MPI PKS))

IH and Lukas Janssen, Phys. Rev. Lett. 113, 106401 (2014)

ERG2014, Lefkada Island

Quadratic band crossing in 2D: ( e. g. bilayer graphene )

Irreducible Hamiltonian: ( )

With short-range interaction:

has an instability at weak coupling:

towards QAH (gapped) ( ) or nematic (gapless) phase. (Sun et al, 2010, Dora, IH, Moessner, 2014)

Three dimensions: gapless semiconductors (gray tin, HgTe,…)

Luttinger spin-orbit Hamiltonian (J= 3/2) (Luttinger, PR 1956)

with (twice degenerate) eigenvalues: (with full rotational symmetry)

Density of states now vanishes at the QTP: short-range interactions are irrelevant, but there is no screening.

What is the effect of long-range Coulomb interaction?

Without the hole band empty, at ``zero” (low) density:

Wigner crystal !

With the hole band filled and particle band empty: the system is

``critical”

In the RG language, changing the cutoff causes the charge to ``flow”

with the dynamical critical exponent:

(Coulomb interaction . ) (Abrikosov, JETP 1974)

Below and near the upper critical dimension, dup = 4, the system is in the non-Fermi liquid interacting phase, with the charge at the fixed point value:

with the small parameter

and the dynamical critical exponent Z < 2.

This implies power-laws in various responses, such as specific heat:

(Abrikosov, JETP 1974, Moon, Xu, Kim, Balents, PRL 2014)

Easy way to get a NFL phase in 3D!

Or not?

The picture must somehow break down before the dimension reaches d = 2; a short range coupling flows like

+ high. ord. term.

and becomes marginal in d=2.

What can happen to the NFL stable fixed point?

The mechanism : collision of UV and IR fixed points (Kaveh, IH, 2005, Gies, Jaeckel 2006, Kaplan, Lee, Son, Stephanov, 2009). First we rewrite the Luttinger Hamiltonian as :

where, (Abrikosov, JETP 1974, Murakami et al, PRB 2004)

and the Dirac matrices satisfy:

The full interacting theory, with long-range and short-range interactions is then: (IH and Lukas Janssen, PRL 2014)

and appears (but actually is not!) O(5) symmetric. Change of the cutoff now amounts to (to one loop)

in addition to:

( Here, the dimensionless charge is defined as: )

Without the long-range interaction (e=0), the theory possesses a quantum critical point (QCP0); weakly coupled close to d=2:

Close to and below d=4 there is a (IR stable) NFL fixed point, but also a (UV stable) quantum critical point at strong interaction: (d=3.5)

They get closer, but remain separated in the coupling space!

At some “lower critical dimension” NFL and QCP collide:

In one loop calculation, this occurs at , and thus above, but close to three dimensions.

Finally, below dl the NFL and QCP become complex, and there is only a runaway flow left:

The system is unstable.

The instability, and the nature of the QCP:

At d=dl the NFL and QCP merge at (0.002, -0.153). Neglecting g1, the flow of g2 in the large-N theory is :

For N >>1, introducing the order parameter the saddle point at

is an exact solution. The O(5) and the rotational symmetry are therefore broken at

which is precisely the RG fixed point.

At large negative g2 the system should develop anisotropic gap and,

The gap is minimal at the equator (in momentum space) at

and the system looks as if under strain. The resulting ground state:

(topological) Mott insulator

( IH and Janssen, PRL 2014 )

The state is equivalent in symmetry to ``uniaxial nematic”.

The fate of NFL: if dl is above but close to d=3, the flow becomes slow close to (complex!) NFL fixed point. The RG ``escape time” is long:

with non-universal constants C and B. There is wide crossover region of the NFL behavior within the temperature window

with the critical temperature,

And the characteristic energy scale for interaction effects as

Assuming a small band mass

and a high dielectric constant

still gives a reasonable

and a detectable

Conclusion:

1) Abrikosov’s non-Fermi liquid phase at T=0 exists only in dimensions d:

dlow < d < dup = 4

with lower critical dimension dlow > 2, and probably close to three.

2) Below dlow the system develops a gap, and most likely becomes a (topological) Mott insulator. (The other possibility is a s-wave superconductor, with an isotropic gap.)

3) NFL shows up in a possibly wide crossover regime of energy scales.

4) Gray tin or mercury telluride should be a (topological) Mott insulator at T=0, and at zero doping!